**2.5.1 Pseudoplastic fluids**

Pseudoplastic fluids become thinner when the shear rate increases, until the viscosity reaches a plateau of limit viscosity. This behaviour is caused by increasing the shear rate and the elements suspended in the fluid will follow the direction of the current. There will be a deformation of fluid structures involving a breaking of aggregates at a certain shear rate and this will cause a limit in viscosity. For pseudoplastic fluids the viscosity is not affected by the amount of time the shear stress is applied as these fluids are non-memory materials i.e. once the force is applied and the structure is affected, the material will not recover its previous structure (Schramm, 2000). Some examples are corn syrup and ketchup.

#### **2.5.2 Viscoplastic fluids**

Viscoplastic fluids, such as e.g. hydrocarbon greases, several asphalts and bitumen, behave as pseudoplastic fluids upon yield stress. They need a predetermined shear stress in order to

Yield stress (0) is defined as the force a fluid must be exposed to in order to start flowing. It reflects the resistance of the fluid structure to deformation or breakdown. Rheograms from rotational viscometer measurements are used as a means to calculate yield stress. It can also be obtained by applying rheological mathematical models (section 2.6; Spinosa & Loito 2003). Yield stress is important to consider when mixing reactor materials, since the yield stress is affecting the physico-chemical characteristics of the fluid and impede flow even at relative low stresses. This might lead to problems like bulking or uneven distribution of

The static yield stress (s) is the yield stress measured in an undisturbed fluid while dynamic yield stress is the shear stress a fluid must be exposed to in order to become liquid and start flowing. The fact that both dynamic yield stress and static yield stress sometimes may appear is explained by the existence of two different structures of a fluid. One structure is not receptive to the shear stress and tolerates the dynamic yield stress, while a second structure (a weak gel structure) is built up after the fluid has been resting a certain period of time (Yang *et al*., 2009). When these two structures merge, a greater resistance to flow is

The formation of the weak gel structure may be a result from chemical interactions among polysaccharides or between proteins and polysaccharides (Yang *et al*., 2009). The weak gel

Non-Newtonian fluids do not show a linear relationship between shear stress and shear rate. This is due to the complex structure and deformation effects exhibited by the materials involved in such fluids. The non-Newtonian fluids are however diverse and can be characterised as e.g. pseudoplastic, viscoplastic, dilatant and thixotropic fluids (Schramm,

Pseudoplastic fluids become thinner when the shear rate increases, until the viscosity reaches a plateau of limit viscosity. This behaviour is caused by increasing the shear rate and the elements suspended in the fluid will follow the direction of the current. There will be a deformation of fluid structures involving a breaking of aggregates at a certain shear rate and this will cause a limit in viscosity. For pseudoplastic fluids the viscosity is not affected by the amount of time the shear stress is applied as these fluids are non-memory materials i.e. once the force is applied and the structure is affected, the material will not recover its

Viscoplastic fluids, such as e.g. hydrocarbon greases, several asphalts and bitumen, behave as pseudoplastic fluids upon yield stress. They need a predetermined shear stress in order to

previous structure (Schramm, 2000). Some examples are corn syrup and ketchup.

structure is quite vulnerable and, thus easily interrupted by increasing shear rates.

**2.3 Dynamic yield stress** 

material in a reactor (Foster, 2002).

generated translated to the static yield stress.

**2.4 Static yield stress** 

**2.5 Non-Newtonian fluids** 

**2.5.1 Pseudoplastic fluids** 

**2.5.2 Viscoplastic fluids** 

2000).

start flowing. One type of these, the Bingham plastic, requires the shear stress to exceed a minimum yield stress value in order to go from high viscosity to low viscosity. After this change a linear relationship between the shear stress and the shear rate will prevail (Ryan, 2003). Examples of Bingham plastic liquids are blood and some sewage sludge's.

#### **2.5.3 Dilatant fluids**

Dilatant fluids become thicker when agitated, i.e. the viscosity increases proportionally with the increase of the shear rate. Like for the pseudoplastic fluids the stress duration has no influence, i.e. when the material is disturbed or the structure destroyed it will not go back to its previous state. Some examples of shear thickening behaviour are honey, cement and ceramic suspensions.

#### **2.5.4 Thixotropic fluids**

Thixotropic fluids are generally dispersions, which when they are at rest construct an intermolecular system of forces and turn the fluid into a solid, thus, increasing the viscosity. In order to overcome these forces and make the fluid turn into a liquid and which may flow, an external energy strong enough to break the binding forces is needed. Thus, as above a yield stress is needed. Once the structures are broken, the viscosity is reduced when stirred until it receives its lowest possible value for a constant shear rate (Schramm, 2000). In opposite to pseudoplastic and dilatant fluids, the viscosity of thixotrpic fluids is time dependent: once the stirring has ended and the fluid is at rest, the structure will be rebuilt. This will inform about the fluid possibilities of being reconstructed. Wastewater and sewage sludge can be examples of fluids with thixotropic behaviour (Seyssieq & Ferasse, 2003) as well as paints and soap.

#### **2.6 Rheological mathematical models**

There are several rheological mathematical models applied on rheograms in order to transform them to information on fluid rheological behaviour. For non-Newtonian fluids the three models presented below are mostly applied (Seyssiecq & Ferasse, 2003).

#### **2.6.1 Herschel Bulkley model**

The Herschel Bulkley model is applied on fluids with a non linear behaviour and yield stress. It is considered as a precise model since its equation has three adjustable parameters, providing data (Pevere & Guibaud, 2006). The Herschel Bulkley model is expressed in equation 5, where 0 represents the yield stress.

$$\mathbf{T} = \mathbf{\tau}\_0 + \mathbf{K} \; \mathbf{\ast} \; \mathbf{\ast}^m \; \tag{5}$$

The consistency index parameter () gives an idea of the viscosity of the fluid. However, to be able to compare -values for different fluids they should have similar flow behaviour index (n). When the flow behaviour index is close to 1 the fluid´s behaviour tends to pass from a shear thinning to a shear thickening fluid. When n is above 1, the fluid acts as a shear thickening fluid. According to Seyssiecq and Ferasse (2003) equation 5 gives fluid behaviour information as follows:

A rotational rheometer RheolabQC coupled with Rheoplus software (Anton Paar) was used for different reactor fluids, which recorded the rheograms´ and allowed subsequent data analysis. The temperature was maintained constant at 370.2 °C. The reactor fluid volume used for each measurement was 17 ml. Reactor fluids from mesophilic (37°C) lab-scale reactors (4 L running volume), with a hydraulic retention time (HRT) of 20 days, were

Five lab-scale reactors (A-E) were sampled before the daily feeding of substrates. All reactors had been running for at least three HRTs prior to sampling. The different substrates treated were slaughter household waste, biosludge from pulp- and paper mill industries, wheat stillage and cereal residues. The TS values ranged between 3.1−3.9 % for four of the

Reactor Digested substrate TS (%) A Slaughter house waste 3.9 B Biosludge from pulp- and paper mill industry 1 3.8 C Biosludge from pulp- and paper mill industry 2 3.7 D Wheat stillage 3.0 E Cereal residues 7.7 Table 1. Fluids from five lab-scale reactors were chosen for rheological measurements. A

Rheological measurements were carried out with a three-step protocol where (1) the shear rate increased linearly from 0 to 800 s-1 in 800 sec., (2) maintaining constant shear rate at 800 s-1 in 30 sec, (3) decreasing linearly the shear rate from 800 to 0 s-1 in 800 sec., according to Björn *et al*. (2010). For each sample three measurements were carried out and performed

The fluid behaviour was interpreted by the flow- and viscosity curves according to Schramm (2000), and the dynamic viscosity, limit viscosity and yield stress were noticed. The three most common mathematical models for non-Newtonian fluids; Herschel Bulkley model; Ostwald model (Power Law) and Bingham model, were applied in order to transform rheogram data values to the rheological behaviour of the fluids. Flow behaviour

The flow curves for reactor fluids A-E (Figures 2−3) indicated different flow behaviour according to the definitions by Schramm (2000). A Newtonian behaviour of reactors A and D, fed with slaughter house waste and wheat stillage, respectively, was illustrated where the exerted shear stress was almost proportional to the induced shear rate. However, a small yield stress of 0.2 Pa and 0.3 Pa were detected, indicating a pseudo-

Fluids from reactor B, receiving biosludge from paper mill industry 1 as substrate, indicated an unusual performance at the beginning of the rheogram with decreasing shear stress, thereafter a linear increase in shear stress. A yield stress of 14 Pa was detected. A space

**3.1 Rheological measurments** 

reactors while one was at 7.7 % (Table 1).

short description of their TS values and substrates are presented.

immediately after sampling or stored at +4 ºC pending analysis.

index (n) and consistency index (K) were studied.

**3.2 Flow and viscosity behaviour characteristics** 

Newtonian behaviour.

sampled.

0 = 0 & n = 1 Newtonian behaviour 0 > 0 & n = 1 Bingham plastic behaviour 0 = 0 & n < 1 Pseudoplastic behaviour 0 = 0 & n > 1 Dilatant behaviour

#### **2.6.2 Ostwald model**

The Ostwald model (Eq. 6), also known as the Power Law model, is applied to shear thinning fluids which do not present a yield stress (Pevere *et al*., 2006). The n-value in equation 6 gives fluid behaviour information according to:

$$\mathbf{r} = \mathbf{K} \, \ast \,\, \mathbf{y}^{(\mathbf{n} \cdot \mathbf{1})} \tag{6}$$

n < 1 Pseudoplastic behaviour n = 1 Newtonian behaviour n > 1 Dilatant behaviour
